Lung function analysis method and apparatus

ABSTRACT

A method for generating data indicative of lung function of a subject. The method comprises receiving first data which has been obtained from the subject, and inputting said first data to a model of lung function to generate said data indicative of lung function. The model of lung function comprises a first model component modelling transfer of gaseous oxygen from a gaseous space within the lung to biological material within the lung based upon quantitative data indicative of oxygen content in the inhaled gases and oxygen content in the biological material and a second model component modelling the transfer of oxygen from the lungs by oxygenation of venous blood to create oxygenated blood based upon quantitative data indicative of oxygen content in the venous blood.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a national phase application of PCT Application No.PCT/GB2010/001989, filed Oct. 27, 2010 which claims priority to UnitedKingdom Patent Application No. 0919269.1, filed Nov. 3, 2009, the entirecontents of which are hereby incorporated by reference therein.

FIELD

The present invention relates to methods for generating data indicativeof lung function and more particularly but not exclusively to theapplication of a mathematical model to data produced by oxygen-enhancedmagnetic resonance imaging of a lung in order to generate further dataindicative of lung function.

BACKGROUND

It will be appreciated that a clinician will wish to test lung functionfor a number of reasons. For example, it can be informative tocharacterise lung ventilation because such ventilation can be affectedby a range of pulmonary disorders. Currently, standard lung functiontests can assess a wide range of global variables describing lungphysiology but cannot be used to investigate disease regionally withinthe lung.

Scintigraphy allows for investigation within a particular region of alung by producing images of a radiation emitting material as it passesthrough a subject. The technique generally necessitates the inhalationof radioactive substances and is limited by low spatial resolution.

One particular example of a valuable scintigraphic imaging technique isa ventilation/perfusion (V/Q) scan, which comprises two imaging steps.First, the subject is asked to breathe a gas mixture containing a gammaradiation emitter, such as radionuclide xenon or technetium, whileimages of the subject's chest are recorded using a gamma camera. Theresulting images show light areas in lung regions to which the gas haspermeated and therefore represent information about which areas of thelungs are being ventilated and which are not. Second, an intravenousinjection of a further gamma emitter, such as radioactive technetiummacro aggregated albumin (Tc99m-MAA), is administered to the subjectwhile recording further images of the subject's chest using the gammacamera. The resulting images show light areas wherever the gamma emitterhas been carried within the lungs by the blood stream and thereforerepresent information about which regions of the lungs are being wellperfused with blood. It will be appreciated that in some cases theresulting images are seen in negative, i.e. dark areas indicate higherconcentrations of radiation emitting material as opposed to light areasas mentioned above.

The clinician, analysing the results of a V/Q scan may be interested inphysiological parameters such as how well certain regions of the lungsare being ventilated with air or perfused with blood, but is oftenparticularly interested in mismatches between the quality of ventilationand of perfusion in local regions in the lungs. Such mismatches indicatethat gas transfer between the air and blood in the lungs is inefficient,because lung regions supplied with inhaled air are not being suppliedwith blood and/or lung regions supplied with blood are not beingsupplied with air. In the case of both chronic illnesses such asobstructive pulmonary disease and acute illnesses such as pulmonaryembolism, V/Q scans are particularly valuable since they can be used toidentify obstructions within the airways or blood vessels which arecausing a ventilation/perfusion mismatch in a region of a lung.Unfortunately, V/Q scans provide relatively low resolution, twodimensional results (i.e. images). It will be appreciated that morelocal information, such as would be provided by volumetric data, wouldbe extremely valuable in order to provide more detailed prognoses andtargeted therapies for addressing ventilation/perfusion problems. Higherresolution imaging technologies, such as magnetic resonance imaging(MRI) or x-ray computed tomography (CT), have therefore beeninvestigated for their applicability to imaging the lungs.

Oxygen-enhanced magnetic resonance imaging (OE-MRI) is a high resolutionimaging technology which has been demonstrated in both healthyvolunteers and patients with pulmonary disease as an alternative,indirect method to visualize lung ventilation. Molecular oxygen isparamagnetic and so acts as an MRI contrast agent when dissolved inwater due to its effect on T₁ (which is known to those skilled in theart of MRI as the longitudinal relaxation time). Another standard outputparameter of MRI is R₁, which is the rate of longitudinal relaxation andis therefore derivable from T₁ as R₁=1/T₁.

Oxygen is always present in a living animal body so it is not possibleto use oxygen as a contrast medium in the usual way (i.e. by assumingthat all visible contrast is that which has been introduced for thepurposes of the study, as is the case with nuclear medicine studies).Rather, at least one baseline measurement is made while oxygen in thelung tissues is at a first concentration, and at least one contrastmeasurement is made while the oxygen in the lung tissues is at a secondconcentration. The difference between the baseline measurement(s) andthe contrast measurement(s) is then calculated. Breathing 100% oxygenresults in an increase in the concentration of dissolved oxygen in thewater contained within both lung tissue and blood within the lungs whencompared to breathing room air (at 21% oxygen), and this can be observedin data indicating the difference between the baseline measurements andthe contrast measurements. More particularly, increased concentration ofdissolved oxygen produces a corresponding decrease in T₁ which can bedetected as a regional signal intensity increase in a T₁-weighted image,denoted here as ΔT₁ (thus the difference in R₁ as described above isdenoted ΔR₁). The resulting data, ΔR₁, represents the increase indissolved oxygen concentration in the lungs.

It is possible to calculate the change in partial pressure of oxygen ina region of interest within a subject from the change in longitudinalrelaxation rate ΔR₁ determined by OE-MRI. An approximate averageconversion factor r₁ between a value of ΔR₁ and a value representing thepartial pressure of oxygen is r₁=2.49×10⁻⁴ s⁻¹mmHg⁻¹ (Zaharchuk G,Martin A J, Dillon W P. “Noninvasive imaging of quantitative cerebralblood flow changes during 100% oxygen inhalation using arterialspin-labeling MR imaging.” American Journal of Neuroradiology. April2008;29(4):663-7). This factor can be used to convert ΔR₁ measurementsto partial pressures of oxygen (in units of mmHg) by dividing each ofthe ΔR₁ values generated by an OE-MRI study by the conversion factor r₁,i.e. ΔPW_(O2)=ΔR₁/r₁, where ΔPW_(O2) is the change in partial pressureof oxygen in units of millimeters of mercury (mmHg) in tissue water. Itis known to perform a plurality of scans, i.e. a “study”, on the samepatient over a short time scale so as to produce a dataset whichrepresents the change in partial pressure of oxygen in a number of localregions of interest within the subject's lungs over the period of thestudy.

High resolution analysis of the lungs by multiple OE-MRI scans over timeis made difficult by the change in size and shape of the lungs from oneimage to the next due to breathing. Breath-holding has been used in somestudies but in patients with lung disease this can be uncomfortable and,as a result, difficult to perform in a reproducible manner. It may alsobe argued that breath-holding interferes with the phenomena beingassessed since it requires large static inhalations which may lead tospurious interpretation of normal breathing function. Accordingly imageregistration methods have been developed to correct for breathing motionbefore calculation of ΔR₁ (e.g. see Naish et al. (2005) MagneticResonance in Medicine 54:464-469). Such methods allow registration of alung outline such that data resulting from multiple scans of the samesubject over time can be registered together with the result that datavalues relating to corresponding locations within the subject's lungsare identifiable as such. Moreover, registration of datasets betweendifferent subjects may allow comparison of lung function between thesubjects in such a way that differing lung sizes and shapes are at leastpartially accounted for.

Each data value produced by an OE-MRI scan as described above relates tothe change in partial pressure of oxygen for a single point in time.Even when a study is performed, i.e. a plurality of scans over a giventime period, the results are a number of scalar data values for aplurality of time points. It is not therefore possible to directlymeasure desired aspects function of the lung, using such a method. If amethod could be found to utilise the data values produced by an OE-MRIstudy so as to infer functional information about a subject thenfunctional data could be determined at OE-MRI resolutions over theregion of the lungs. This data would be extremely useful in diagnosisand prognostic estimation for people or animals.

An existing method of generating functional data about the lungs fromOE-MRI data is to begin by constructing a simple, highly abstractedfunctional model comprising model parameters the values of which areexpected to change in response to a change in lung function. The modelis then “fitted” to data generated for a subject over time for a regionof interest. The fitting can take many forms, but its general purpose isto apply the model to the data so as to determine values for one or moremodel parameters which result in the model representing the data withacceptable accuracy. The values of the model parameters which have beendetermined by the fitting are then used to compare lung function betweensubjects or between datasets acquired from the same subject over time.

Such a model is validated by gathering empirical evidence that, oncefitted to data, the model provides a set of data values which havediagnostic capability. That is, the value of at least one parameter ofthe model changes depending on the health and/or particular diagnosis ofthe subject. The values produced by such a model are not, however,suitable for quantitative analysis of lung function because none of themodel parameters relate directly to measurable physiological parameters.Rather, the model parameters are abstract indicators which have beenarrived at by roughly approximating physiological function.

SUMMARY

It is an object of the present invention to obviate or mitigate at leastone of the problems described above.

According to a first aspect of the invention there is provided a methodfor generating data indicative of lung function of a subject, the methodcomprising receiving first data which has been obtained from the subjectand inputting said first data to a model of lung function to generatesaid data indicative of lung function; wherein the model of lungfunction comprises a first model component modelling transfer of gaseousoxygen from a gaseous space within the lung to biological materialwithin the lung based upon quantitative data indicative of oxygencontent in the inhaled gases and oxygen content in the biologicalmaterial, and a second model component modelling the transfer of oxygenfrom the lungs by oxygenation of venous blood to create oxygenated bloodbased upon quantitative data indicative of oxygen content in the venousblood.

The method according to the first aspect of the present invention isadvantageous in that the first and second model components provideimproved model accuracy by considering both the oxygen content in thebiological material in the lungs and in the venous blood respectively.It has been realised that the consideration of these two particularconcentrations of oxygen in the model allows sufficiently accuratemodelling of lung function so as to produce data indicative of lungfunction.

Biological material in the context of the invention means thenon-gaseous matter which makes up the lung. More particularly,biological material includes blood and tissues, whether lung tissues (orparenchyma), blood vessels or other tissues including cancerous tissue.Biological material may also include non-blood body fluids. Gaseousspaces and alveolar spaces in the context of the invention meannon-solid and non-liquid spaces within the lungs. More particularly,gaseous spaces and alveolar spaces include bronchi, bronchioli, alveoli,alveolar ducts and alveolar sacs within the lungs.

The quantity of data indicative of oxygen content in the inhaled gasesmay be indicative of a change of oxygen content in the inhaled gases.The quantitative data indicative of oxygen content in the biologicalmaterial may be indicative of a change of oxygen content in thebiological material. The quantitative data indicative of oxygen contentin the venous blood may be indicative of a change of oxygen content inthe venous blood.

The second model component may comprise a first parameter representing avolume of blood flow. The generated data may include at least one valuefor said first parameter representing a volume of blood flow. The use ofthis first parameter is advantageous in that the model can eitherreceive data which has been determined about blood flow in the subjectas an input to the model or, preferably, can produce quantitative dataabout blood flow from the model. The first model parameter can beexpressed in units of volume of blood per volume of lung per unit oftime.

The first model component may comprise a second parameter representing avolume of inhaled gases in at least part of the gaseous space. Thegenerated data may include at least one value for said second parameterrepresenting a volume of inhaled gases in at least part of the gaseousspace. The use of this second parameter is advantageous in that themodel can either receive data which has been determined about the volumeof gas inhaled by the subject as an input to the model or can producequantitative data about volumes of inhaled gas within the lung from themodel.

The method according to the first aspect of the present invention mayfurther comprise generating data based upon said first parameter andsecond parameter. Generating data based upon said first parameter andsaid second parameter may comprise performing an arithmetic operation ona value of said first parameter and a value of said second parameter.The arithmetic operation may be a division of a ventilation value basedupon said second parameter by a perfusion value based upon said firstparameter.

As described above, information about ventilation and perfusion, andparticularly ventilation/perfusion mismatch, is extremely valuable in aclinical setting. An advantage of a model which is based upon themovement of blood and air within the lungs (rather than the movement ofa radiation emitting contrast medium) is that the model is capable ofproducing quantitative values which can be compared against knownstatistical averages, compared between patients or compared betweenmultiple scans of the same patient over time. Known V/Q scans are notsuitable for this kind of quantitative comparison due to theunpredictability of the movement of the contrast. Rather, the imageswhich result from known V/Q scans are assessed or compared visually by aclinician.

The model of lung function may model lung function in a part of a lungcomprising a first portion comprising said gaseous space and a secondportion comprising said biological material. The first parameter mayrepresent a volume of blood flow thorough said second portion of thepart of the lung. The second parameter may represent the volume ofinhaled gasses in the first portion of the part of the lung.

The quantitative data indicative of oxygen content in the venous bloodmay comprise a plurality of values each relating to oxygen content at arespective time. The method may further comprise receiving thequantitative data indicative of oxygen content (or change in oxygencontent) in the venous blood, the quantitative data indicative of oxygencontent in the venous blood having been obtained from the subject. Thatis, a test or tests may be carried out on the subject to determinequantitative data indicative of oxygen content in the venous blood.

The model may include a component representing the solubility of oxygenin blood. The component representing the solubility of oxygen in bloodmay be based on a dissociation curve of oxygen-haemoglobin saturation.The model may approximate the solubility of oxygen in the blood using alinear function, or alternatively a quadratic function or a spline.

The first model component may comprise a first part representing theamount of inhaled gases. The first model component may further comprisea second part representing the amount of oxygen diffused into thebiological material from the gaseous space. The first model componentmay be defined by a difference between said first part and said secondpart of said first model component.

The second model component may comprise a first part representing theamount of oxygen diffused into the biological material from the gaseousspace. The second model component may further comprise a second partrepresenting the transfer of oxygen from the lungs by oxygenation ofvenous blood. The second model component may be defined by a differencebetween said first part and said second part of said second modelcomponent.

The model may assume that the concentration of oxygen in the gaseousspace within the lung is substantially proportional, by a known constantof proportionality, to the oxygen concentration in the biologicalmaterial within the lung. The model may assume that the concentration ofoxygen in the gaseous space within the lung is substantially equal tothe concentration of oxygen in the biological material.

The model may comprise a third parameter indicating a proportion of apart of the lung made up of one of said first and second portions. Themethod according to the first aspect of the present invention mayfurther comprise receiving at least one value for the third parameter asan input. The at least one value for the third parameter may have beenobtained from the subject. That is, data (e.g. image data) may beobtained from the subject to generate the value for the third parameter.The data may be obtained by any appropriate method, including by anx-ray computed tomography scan or a magnetic resonance proton densityscan.

The first and second model components may be combined together in themodel according to said proportion. The model may have the form:

$\left. \frac{{PW}_{O_{2}}}{t} \right).$

wherein PW _(O) ₂ is an input parameter representing the partialpressure of oxygen in the biological material in said part of the lung;FI _(O) ₂ is an input parameter representing the concentration of oxygeninhaled by the subject; Cv _(O) ₂ is an input parameter representing theconcentration of oxygen present in the venous system of the subject;α′_(O) ₂ is an input parameter which represents the solubility of oxygenin the biological material; β_(O) ₂ is an input parameter whichrepresents the solubility coefficient of oxygen in the blood which maybe approximated by a linear relationship with respect to the partialpressure of dissolved oxygen PW _(O) ₂ over a typical range of partialpressures observed during a study; ν_(W) is a parameter representing theproportion of biological material in said part of the lung, {dot over(V)}A is an output parameter for the model which represents the volumeof inhaled gases within said gaseous space in said part of the lung; Cois an extrapolation of the concentration of oxygen in blood at zeropartial pressure based upon the linear relationship used to determineβ_(O) ₂ ; and {dot over (Q)} is an output parameter for the model whichrepresents the volume of blood which passes through said part of thelung.

A model of the general type described above may be processed tointegrate the term indicating a rate of change of the partial pressurewith respect to time (i.e. to eliminate the term

$\left. \frac{{Pw}_{O_{2}}}{t} \right).$

This integration may result in the replacement of that term with termsindicating measurements of partial pressure oxygen within the biologicalmaterial at a plurality of time points. This is useful in that dataindicating measurements of the partial pressure of oxygen in thebiological material at a plurality of time points is data which can morereadily be obtained from OE-MRI data. The elimination of the rate ofchange term can be carried out in any convenient way. For example, anintegrating factor may be used to perform the elimination.

One advantage of some embodiments of the invention is the use ofmathematical models that include parameters which correspond to standardphysiological parameters, e.g. of lung function. In particular,mathematical models according to some embodiments of the inventioninclude model parameters for at least one of, or some combination of:the volume of alveolar gas in units of volume of gas per unit of timeper unit volume of lung, the systemic (non-local) venous oxygenconcentration (i.e. the concentration of oxygen in the blood arriving atthe lungs) in units of volume of gas per unit volume of blood, thecapillary blood flow through a particular region of interest in units ofvolume of blood per unit of time per unit volume of lung, and thesolubility of oxygen in blood (whole blood including blood plasma) inunits of volume of oxygen per volume of blood for a given pressure.

A model which incorporates physiologically accurate model parameters isnecessarily complex because the physiology which is being modelled iscomplex. It is difficult to apply such a model to a real situation sincethe model has a large number of unknowns. The inventors have realisedthat it is possible to start with such a complex model and simplify itin such a way that the resulting simplified model remains clinicallysignificant and contains accepted physiological parameters. If one ormore measurements are made of some of the model parameters then theseparameters do not need to be fitted to the data but become inputs to themodel and/or if assumptions are made with regard to certain parametersof the model then these parameters also do not need to be fitted to thedata. Critically, simplification of the complex model in this way usingknown assumptions and clinically acceptable measurements maintains theclinical significance of the remaining modelled parameters. Moreover,any assumptions are made in relation to known physiological parametersand the validity of the assumptions can be verified against theliterature. Therefore, the resulting simplified model is delivered witha known set of validated assumptions which can be used to validate theresults ultimately generated by fitting the simplified model to thedata.

The quantitative data indicative of oxygen content in the biologicalmaterial may be based upon oxygen enhanced magnetic resonance data. Thegeneration of data indicative of lung function in the method accordingto the first aspect of the invention may comprise application of theLevenberg Marquardt non-linear least squares fitting algorithm or otherfitting algorithm to the model so as to fit said model to said inputdata.

According to a second aspect of the invention, there is provided amethod for generating data indicative of ventilation and perfusionwithin a subject's lung the method comprising obtaining first dataindicative of ventilation and perfusion while the subject inhales gasescomprising a first concentration of oxygen; obtaining second dataindicative of ventilation and perfusion while the subject inhales gasescomprising a second concentration of oxygen; generating third data basedupon the first and second data; and determining quantitative dataindicative of ventilation and perfusion based upon the third data.

An advantage of the method according to the second aspect of theinvention is that by obtaining data while the subject inhales twodifferent concentrations of oxygen quantitative data indicative ofventilation and perfusion can be determined, while traditional methodsare limited to determining non-quantitative information aboutventilation and perfusion determined indirectly from the propagation ofcontrast materials which are not normally present within the lungs.

The first concentration of oxygen may be approximately 21%, i.e. theconcentration normally present in atmospheric air, and/or the secondconcentration of oxygen may be approximately 100%. The first and/orsecond data may be magnetic resonance data obtained from the subject. Ingeneral terms, the data obtained according to the second aspect of theinvention may be obtained by an oxygen-enhanced magnetic resonancestudy, the parameters for which may be any desirable parameters forobtaining such data. It will be appreciated that it is not necessary toinstantaneously switch between the different concentrations of oxygenbreathed by the subject (21% and 100% in the above example). Indeed, theonly requirement is that the concentration of oxygen breathed by thesubject varies over time and that the concentration breathed by thesubject at any given time is known.

The determining may comprise processing the third data using amathematical model. The processing may comprise determining at least onevalue of a parameter of the mathematical model based upon said thirddata.

The mathematical model of the second aspect of the present invention maycomprise a first model component modelling transfer of gaseous oxygenfrom a gaseous space within the lung to biological material within thelung based upon quantitative data indicative of oxygen content in theinhaled gases and oxygen content in the biological material and a secondmodel component modelling the transfer of oxygen from the lungs byoxygenation of venous blood to create oxygenated blood based uponquantitative data indicative of oxygen content in the venous blood.

The mathematical model of the second aspect of the invention maycomprise any of the features of the first aspect of the invention.

Subjects tested according to the methods of the invention may be anysubject for which it is desirable to test lung function. The subject maybe a mammal (although the methodology is also generally applicable toany organism with a lung). The subject may be a human.

The methods are particularly useful for testing human subjects withconditions such as pulmonary embolism, asthma, COPD, fibrotic lungdiseases, emphysema, bronchitis, alpha1-antitrypsine deficiency orbronchiectasis; or in the case of airway constriction or alveolar damagecaused by smoking or environmental factors. Such human subjects areknown to have impaired lung function and providing an indication of theextent of the impairment in each of a plurality of local regions withinthe lung is a valuable measurement which may determine the approach totreatment of each human subject.

It is preferred that an OE-MRI study is used to gather a plurality ofdata values about oxygen partial pressure changes from various regionsof interest within the lungs of a subject. It will of course beappreciated that the applicability of the mathematical model is notlimited to use with OE-MRI data and may be combined with other imagingmethodologies and/or diagnostic medical tests. In the preferredembodiment using OE-MRI, subjects to be tested should be placed in anMRI machine typically but not necessarily at 1.5 tesla magnetic fieldstrength. As the method requires little specialist equipment it shouldbe possible to use OE-MRI in any MRI machine designed for human oranimal use. A T₁-weighted imaging protocol should be chosen which issuitable for lung imaging, i.e. which can overcome the problems causedby low proton density in the lung and the magnetic field in homogeneityinduced by the many air-tissue interfaces of the lung, and one which isalso sufficiently sensitive to the signal changes induced by changes ininhaled oxygen concentration, e.g. an Inversion Recovery Half FourierSingle-Shot Turbo Spin-Echo (IR-HASTE) sequence, or an InversionRecovery Snapshot Fast Low Angle-Shot (IR Snapshot FLASH) sequence.Gases are typically delivered at a rate of 10-15 l/min.

In one embodiment, during a study, the subject inhales gases with atleast two different partial pressures of a paramagnetic gas (e.g.oxygen), and may be fitted with a mask or breathing apparatus for gasdelivery in order that different gases may be inhaled while the MRIscans are performed. When the gas is oxygen room air may be used as oneof the partial pressures of oxygen in which instance the subject wouldbreathe normally without the use of any apparatus. In some embodimentsmedical air, which mimics room air but is administered from a controlledsource is used as it is easier to switch between one administered gasand another than to switch between a non-administered source (i.e.normally breathed room air) and an administered source.

It is preferred that the subject inhales two gases—a first gas whichcomprises a relatively low concentration of oxygen (e.g. 10%-35%) and asecond gas which comprises a relatively high concentration of oxygen(e.g. 45%-100%). It is most preferred that the first gas is air(comprising approximately 21% oxygen) and the other is a gas comprisingan oxygen content of 90%-100%. It will be appreciated that the choice ofgases used may depend on the health status of the subject.

The subject may revert back to breathing the first gas. In this event,measurements are preferably made which detect the change inconcentration of dissolved oxygen within the lungs during this furthertransition period. Transitions between each gas may be repeated asneeded.

It is preferred that OE-MRI data is recorded for each region of interestwithin the lungs by starting a subject on a low concentration of oxygenthen swapping the inhaled gas to one with a high oxygen concentrationfor a period of time. The method of the invention most preferablygenerates OE-MRI data from a subject when the subject is breathingnormal air (e.g. medical air comprising 21% oxygen) and 100% oxygen. Thediffering concentrations of the oxygen, acting as a contrast medium,then influence the MRI signal detected from protons (primarily from thehydrogen nuclei found in water or lipids in the pulmonary tissue) ifproton MRI is being employed but potentially other MRI-visible nuclei ifnon-proton MRI is being employed, and this OE-MRI data is then used tocreate the input for the model.

In a subject with healthy lung function the Oxygen-Enhanced MRI signalof the lung will have increased and reached saturation withinapproximately 5 min of the subject beginning to inhale 100% oxygen. Thetime for the signal to decrease to its normal baseline value when thegases are switched back to air is also within the same time frame ofapproximately 5 min. Typically the subject will be required to breathe agas mixture or mixtures with a higher concentration of oxygen for amaximum period of approximately 10 minutes. Adverse effects frombreathing higher concentrations of oxygen have only been noted inhealthy subjects after approximately 24 hours exposure, and thereforethis length of exposure is deemed safe and without any detrimentaleffects for the majority of subjects. In patients with serious lungdisorders, such as COPD, the effect of breathing 100% oxygen can bedetrimental due to the effects of oxygen driven breathing, which is aknown complication in illnesses such as COPD. In subjects whosebreathing is driven by oxygen, an increase in the concentration ofoxygen breathed results in a decreased impulse to continue to breatheand can have serious consequences. It will therefore be appreciated thatthe duration of the study is chosen for its balance between subjectcomfort and safety, and data quality. It will further be appreciatedthat, as with all medical studies, some subjects may not be able tosafely undergo the OE-MRI study.

The models according to some embodiments of the invention, onceconstructed, may be solved numerically using any algorithm (such as theLevenberg Marquardt non-linear least squares fitting algorithm) thatallows the fitting of a functional form, as defined by the compartmentalmodel, to the dataset produced by the study, for example, by the OE-MRI.Techniques for fitting functions to measurements are known in the art.

The models according to the invention may be an adaptation of theequations developed by Kety (Kety, S S (1951) Pharmacological Reviews.3: 1-41) who described the rate of diffusion of gases across thealveolus membrane to pulmonary capillary blood.

Throughout the specification, any reference to the volume of gases isgiven at body temperature and pressure, saturated with water vapour(i.e. BTPS).

The aspects of the invention described above can be implemented in anyconvenient way, including by way of methods, apparatus and computerprograms. The invention therefore provides computer programs comprisingcomputer readable instructions which cause a computer to carry out themethods described above. Such computer programs can be carried on anysuitable computer readable medium including tangible media (e.g. disks)and intangible media (e.g. communications signals). The invention alsoprovides apparatus programmed to carry out the methods described above.

It will be appreciated that features described in the context of oneaspect of the invention can be applied to the other aspect of theinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention are now described, by way of example only,with reference to the accompanying drawings, in which:

FIG. 1 is a diagrammatic representation of the transit of oxygen into ahuman body through the lungs;

FIG. 2 is a diagrammatic representation of the components of anembodiment of an imaging technique in accordance with an aspect of thepresent invention;

FIG. 3 is a diagrammatic representation of the relationship between theparameters of an embodiment of a model in accordance with the presentinvention;

FIG. 4 is an annotated graph representing the dissociation curve foroxygen concentration in blood and a linear approximation of that curve;

FIG. 5 (a) shows a sample ventilation/perfusion map generated using amethod according to an embodiment the invention from a healthy subject,and (b) the a similar ventilation/perfusion map for a subject sufferingfrom chronic obstructive pulmonary disease (COPD); and

FIG. 6 (a-c) show histograms of ventilation/perfusion values for threehealthy subjects, and (d-f) show histograms of ventilation/perfusionvalues for three subjects suffering from COPD.

DETAILED DESCRIPTION

FIG. 1 shows a region of interest within a lung 1. Air 2 is inhaled intothe lung 1 and inflates an alveolus 3 within the region of interest asindicated by an arrow A1. Deoxygenated blood 4 flows into the lung 1from the venous system (not shown), as indicated by an arrow A2. Theblood 4 is carried by a capillary 5 in the lung 1 past the alveolus 3,where the deoxygenated blood 4 is oxygenated by gas transfer (indicatedby an arrow A3) between the air 2 in the alveolus 3 and the blood 4 inthe capillary 5 to form oxygenated blood 6. The oxygenated blood 6 isthen transported away from the lung 1 as indicated by an arrow A4. Thealveolus 2 then deflates as the air 2 leaves the alveolus (indicated byan arrow A5) during exhalation of the air 2 from the lung 1.

In order to determine values of at least some of the physiologicalparameters described in relation to FIG. 1 for a particular subject 10,an imaging technique is performed on the subject 10 as depicted in FIG.2. First, the subject 10 is placed inside a magnetic resonance scanner11 and an imaging study is performed while the subject breathes firstair having an oxygen concentration of about 21% and then breathes 100%oxygen.

The study comprises a number of individual scans, each of whichgenerates at least one data value for each of a plurality of regions ofinterest within the lungs of the subject 10. At least one of the scansrelates to the lungs before the subject 10 starts breathing the 100%oxygen. It will be appreciated that, generally, performing more scanswhile the subject breathes the baseline concentration of oxygen (i.e.21%) can improve the imaging technique results because the data producedby those scans for each region of interest are indicative of normal(i.e. non-contrast enhanced) MR signal strengths in the subject and canbe combined, for example by averaging. The regions of interest withinthe lungs are notionally adjacent to one another and arranged in a threedimensional (3D) grid so that the OE-MRI study produces a fourdimensional (4D) dataset (i.e. three spatial dimensions plus time).

The dataset is input to a data preparation module 12. The datapreparation module 12 applies a registration algorithm to the dataset soas to generate a registered dataset in which each spatial location inthe dataset contains a plurality of data values (one for each timepoint) relating to the MR signal during the study at a correspondingspatial location within the lungs. That is, a value of R₁ representingmagnetic resonance longitudinal relaxation rate measured at a givenlocation (x,y,z) in the lung at a given time t may be found in the 4Ddataset at l(x,y,z,t), and a value of R₁ for the same location withinthe lungs for the following time point is given by the locationl(x,y,z,t+1) in the dataset. Thus, for a given location within thelungs, the registered dataset contains a plurality of values (at thesame x,y,z co-ordinates in each dataset) of R₁ for each of a pluralityof time points during the OE-MRI study. The data preparation module 12then subtracts a value of R₁ determined for each regional lung location(x,y,z) when the subject was breathing air having an oxygenconcentration of 21% (the values being determined from the at least onescan taken before the subject started breathing 100% oxygen) from thecorresponding spatial location in the dataset for each time point. TheΔR₁ values in the resulting dataset are then converted to valuesindicative of the change in partial pressure of oxygen using aconversion factor, such as r₁=2.49×10⁻⁴ s⁻¹mmHg⁻¹ as determined byZaharchuk, Martin and Dillon as described above. Accordingly, the outputfrom the data preparation module is a 4D dataset containing data valuesrepresenting the increased dissolved oxygen concentration in each of aplurality of adjacent regions of interest within the lungs over time.Gaseous oxygen within the alveolar spaces in the lungs, such as that inthe air in the alveolus 3, is not visible to OE-MRI. This is because itis the effect on water in which the oxygen is dissolved, and not theoxygen itself, which is detected by OE-MRI. The values of change inpartial pressure data values in the 4D dataset are thereforerepresentative of the change in partial pressure only of dissolvedoxygen, and not gaseous oxygen, in the region of interest.

Data values l(x,y,z,t) for all times t from the dataset which correspondto a given spatial location (x,y,z) within the lungs are input to amathematical model 13, described in detail further below. Othermeasurements or tests 14 may be performed upon the subject 10 so as todetermine other data values which may be input to the model 13. Themeasurements 14 may be of a physiological parameter which does not (oris not deemed to) change over time, or may change over time andtherefore be measured for each of a plurality of time pointscorresponding to the time points of the scans of the OE-MRI studydescribed above. Other inputs may be made to the model by makingestimations 15 of the values of certain parameters for each of the timepoints. It will be appreciated that in some embodiments the furthermeasurements 14 are not needed. Once all data value inputs have been fedinto the model, a fitting operation is performed in which values 16 aredetermined for any remaining model parameters (i.e. unknowns). Thevalues 16 determined for the remaining model parameters are the outputfrom the imaging technique and represent physiological parameters of thelung function of the subject.

The nature of the mathematical model 13, into which the input datavalues are fed and from which output values 16 are generated, isdeterminative to the usability of the output values. A model containingmodel parameters the values of which can be related to, and verifiedagainst, known values of related physiological parameters may be used toquantitatively assess lung function in the subject. In contrast, a modelwithout such verifiable parameters may not be used in this way (althoughsuch a model may remain useful in distinguishing between healthy andunhealthy lung function). The following description relates to animplementation of the mathematical model 13 which may readily be used inrelation to the above imaging technique

In general terms the mathematical model (or models) 13 provides termswhich represent at least some of the parameters of the spaces and thetransfers of oxygen between the spaces depicted in FIG. 1. In each ofthe spaces within the lung 1 the concentration of oxygen is not uniform;by which it is meant that, for example, the air 2 in the alveolus 3 maynot have a uniform distribution of oxygen during inhalation, diffusionor exhalation. Rather the oxygen concentration fluctuates across thespace. However, as shown in FIG. 3, the spaces in the lung shown in FIG.1 can be modelled as a system in which it is assumed that, within eachof the spaces, there is a uniform oxygen concentration. For example, PW_(O) ₂ is the partial pressure of oxygen dissolved in the blood andtissues in a region of interest, which is assumed to be uniformthroughout the blood and tissues within a particular region of interest.It will be noted that all of the transfers indicated by arrows in FIG. 1are replicated by correspondingly labelled arrows in the schematicillustration of the model in FIG. 3. It will of course be appreciatedthat not all of the transfers need to be considered by a model inaccordance with the invention in order for the model to producemeaningful data indicative of lung function.

In any given time period the net input of gaseous oxygen (which, asexplained above, is not OE-MRI visible) into the alveolar spaces 3 canbe modelled as the difference in oxygen concentration between gasentering the alveolar spaces (denoted by the arrow A1) and gas leavingthe alveolar spaces (denoted by the arrow A5), taking into account thedifference in the volume of gas entering and leaving the alveolarspaces, in other words:

dİ _(O) ₂ ={dot over (V)} I F I _(O) ₂ −{dot over (V)} A F A _(O) ₂  (1)

where dİ_(O) ₂ is a numerical value which represents the net volume ofoxygen input into the alveolar spaces within the lungs in units of mlO₂/min/ml lungs; {dot over (V)}I is a value which represents the volumeof inspired oxygen entering the alveolar spaces in units of mlgas/min/ml lungs; {dot over (V)}A is a value which represents the volumeof expired oxygen leaving the alveolar spaces in units of ml gas/min/mllungs; Fİ_(O) ₂ is a value between 0 and 1 which represents thefractional concentration of oxygen in the inspired gas; and FA _(O) ₂ isa value between 0 and 1 which represents the fractional concentration ofoxygen in the expired gas. {dot over (V)}I, {dot over (V)}A, FI _(O) ₂and FA _(O) ₂ are shown in FIG. 3.

Although in reality a subject breathes out a small amount less than isbreathed in, it is reasonable to assume that the volumes inspired andexpired (i.e. {dot over (V)}I and {dot over (V)}A indicated by arrows A1and A5 respectively) are equal. Accordingly:

dİ _(O) ₂ ={dot over (V)} A(F I _(O) ₂ −F A _(O) ₂ )  (2)

It will be appreciated that this assumption can be made more general byexpressing {dot over (V)}I in terms of multiples of {dot over (V)}Arather than by restricting the relationship to an equality.

Oxygen within the alveoli is diffused (as indicated by the arrow A3)into water within nearby tissues 2, and blood 6 in the nearby capillary5 (tissue, blood vessels and blood within the lungs are generallyreferred to herein as biological material). At this point the oxygenbecomes MR visible since it is now dissolved in water. Some of theoxygen dissolved into this water is immediately bonded to haemoglobin inthe oxygenated blood 6, at which point the bonded oxygen again becomesinvisible to MR. Using Fick's diffusion law, the oxygen input into thedeoxygenated blood 4 and tissue water from the alveolar spaces can bedescribed as:

{dot over (V)} _(O) ₂ =D M _(O) ₂ (P A _(O) ₂ −P W _(O) ₂ )  (3)

where {dot over (V)}_(O) ₂ is a value which represents the volume ofoxygen diffused into the water in units of ml O₂/min/ml lung 1; PA _(O)₂ is a value which represents the partial pressure of oxygen in thealveolar spaces in units of mmHg; and PW _(O) ₂ is a value whichrepresents the partial pressure of oxygen dissolved in the water inunits of mmHg. {dot over (V)}_(O) ₂ , PA _(O) ₂ and PW _(O) ₂ are shownin FIG. 3.

DM _(O2) (denoted in FIG. 3 at the boundary of the alveolar space andthe biological material) is the diffusion coefficient for oxygen intothe water given by:

$\begin{matrix}{{DM}_{O_{2}} = \frac{D_{O_{2}}S\; \alpha_{O_{2}}}{H}} & (4)\end{matrix}$

where D_(O) ₂ is a value which represents the diffusion coefficient forthe volume of oxygen diffused through the membranes which border thealveolar spaces in units of cm²/s; S is the surface area of membranesper unit volume in units of mm/ml lung 1; α_(O) ₂ is the solubilitycoefficient of oxygen in the solvent of the membrane (i.e. the membranewater) in units of ml O₂/ml water; and H is the thickness of themembrane in units of mm.

The change in fractional oxygen concentration in an alveolar space (dFA_(O) ₂ ) can be defined as the input oxygen into the alveolar space dueto breathing as set out in equation 2 less the output oxygen from thealveolar space due to diffusion as set out in equation 3. This is setout in equation 5:

$\begin{matrix}{{v_{A}\frac{{FA}_{O_{2}}}{t}} = {{\overset{.}{V}{A\left( {{FI}_{O_{2}} - {FA}_{O_{2}}} \right)}} - {{DM}_{O_{2}}\left( {{PA}_{O_{2}} - {PW}_{O_{2}}} \right)}}} & (5)\end{matrix}$

where ν_(A) is a value between 0 and 1 which represents the proportionof the whole space (including biological material and alveolar gasspaces) which is taken up by gas space; t is a value which representstime; and the equation is thus a differential which represents thechange in oxygen concentration in the alveolar spaces (such as withinthe alveolus 3) for a change in time t.

So as to create a relationship between the input and output parts ofequation 5, it is desirable to convert the fractional concentration ofinspired oxygen into a partial pressure representing gaseous oxygenpartial pressure in the lung 1. This can be achieved by considering thecomposition of the air 2 in the alveolar spaces in terms of partialpressures. It is known that the air 2 in the alveolar spaces should beat roughly barometric pressure (760 mmHg) and that the air 2 shouldcarry a certain amount of water vapour. Oxygen dissolved in water vapourin the air 2 would not be diffused into blood 4 in the capillary 5 andso it is not desirable to consider this in the calculation of thepartial pressure of oxygen in the space. Equation 6 defines theconversion between fractional concentrations of a gas, such as FA _(O) ₂, and partial pressures which are more readily usable and verifiable ina clinical setting:

$\begin{matrix}{{FA}_{O_{2}} = \frac{{PA}_{O_{2}}}{\left( {{PB} - P_{H_{2}0}} \right)}} & (6)\end{matrix}$

where PB is the barometric (i.e. atmospheric) pressure, which isapproximately 760 mmHg; and P_(H) ₂ _(O) is the partial pressure ofwater vapour, which at body temperatures is known to be approximately 47mmHg.

Substituting equation 6 into equation 5 gives:

$\begin{matrix}{{\frac{v_{A}}{{PB} - P_{H_{2}0}}\frac{{PA}_{O_{2}}}{t}} = {{\overset{.}{V}{A\left( {{FI}_{O_{2}} - \frac{{PA}_{O_{2}}}{{PB} - P_{H_{2}0}}} \right)}} - {{DM}_{O_{2}}\left( {{PA}_{O_{2}} - {PW}_{O_{2}}} \right)}}} & (7)\end{matrix}$

In equation 7, the fractional concentration of oxygen input FI _(O) ₂ isnot converted into partial pressures. This is because the concentrationof oxygen breathed by the subject is often known in fractionalconcentrations (e.g. 21% O₂ for room air). It will be appreciated thatin some circumstances rather than estimating the fractionalconcentration of oxygen breathed by a subject, it is desirable tomeasure the partial pressure inhaled, for example by means of a gasanalyser or the like.

In addition to the model of net input to the alveolar gas spaces frombreathing, as given in equation 7, the net input of oxygen into theblood 4 and tissue water from the gas spaces can be modelled usingstandard physiological parameters as is now described.

Recalling equation 3:

{dot over (V)} _(O) ₂ =D M _(O) ₂ (P A _(O) ₂ −P W _(O) ₂ )  (3)

It is known from equation 3 that the gross input of oxygen into theblood 4 and tissue water from the alveolar gas spaces (at arrow A3) isgiven by Fick's law of diffusion. The gross output of oxygen from thelung 1 is made up from oxygen which is carried away (at arrow A4) fromthe lung 1 in the oxygenated blood 6 by blood flow, and from themetabolic consumption of oxygen in the lung tissues themselves. Relativeto oxygen carried away from the lung 1 in the oxygenated blood 6, themetabolic oxygen consumption in the lung 1 is insignificant and cantherefore be approximated to zero. Thus, the oxygen output from a regionof interest in the lung is given to an acceptable approximation by theamount of oxygen carried away in the oxygenated blood 6, which can beexpressed as a function of blood flow and the difference between theoxygen concentration in deoxygenated blood 4 and oxygenated blood 6, asset out in equation 8:

d{dot over (O)} _(O) ₂ ={dot over (Q)}(Cc′ _(O) ₂ −C v _(O) ₂ )  (8)

where d{dot over (O)}_(O) ₂ is a value representing the amount of oxygenleaving the region of interest in units of ml gas/min/ml lung; {dot over(Q)} is the blood flow through the region of interest in ml blood/min/mllung; C v _(O) ₂ is a value representing the systemic venousconcentration of oxygen (i.e. that throughout the subject's body) inunits of ml gas/ml blood; and Cc′_(O) ₂ is a value representing thefractional oxygen concentration of the oxygenated blood 6 as it leavesthe region of interest in units of ml gas/ml blood. {dot over (Q)}, C v_(O) ₂ , Cc′_(O) ₂ are shown in relation to one another in FIG. 3.

For a given local partial pressure of oxygen, certain concentrations ofoxygen are present in the oxygenated blood (both dissolved in bloodplasma and bonded to haemoglobin) and in the lung tissues (i.e. in thetissue water). There is a standard solubility coefficient α_(O) ₂ foroxygen in water. Multiplying the local partial pressure of oxygen bythis coefficient gives the concentration of oxygen in the water in unitsof ml O₂/ml blood.

The concentration of oxygen Cc′_(O) ₂ is determined by a dissociationcurve 17 such as that shown in FIG. 4, determined empirically by Kelman(JAP 21(4): 1375-6, 1966). It can be seen that the local partialpressure of oxygen is indicated on the x-axis and the concentration ofoxygen in the blood is indicated on the y-axis. The portion of thedissociation curve near saturation (above around 20 ml O₂/ml blood),marked 18, is almost straight. The dissociation curve is approximatedusing a straight line 19 (represented in FIG. 4 by a dashed line). Thestraight line is defined by two constants, constant CO indicating apoint at which the straight line 19 meets the y-axis and a gradient ofthe straight line (i.e. the solubility coefficient β_(O) ₂ ). It will beappreciated that different straight line approximations (defined byrespective values of CO and β_(O) ₂ ) may be applied in differentregions of a lung.

A value of the concentration of oxygen in the blood may be approximatedfor partial pressures using only the two constants CO and β_(O) ₂ and avalue indicating the local partial pressure of oxygen, in this case thepartial pressure of dissolved oxygen PW _(O) ₂ , as set out in equation9:

Cc′ _(O) ₂ =C O+β_(O) ₂ P W _(O) ₂   (9)

where Cc′_(O) ₂ is the concentration of oxygen in the blood in ml O₂/mlblood; CO is an offset as described above the concentration of oxygenfor a zero partial pressure PW _(O) ₂ in units of ml O₂/ml blood; andβ_(O) ₂ is a value representing the solubility coefficient of oxygen inthe blood in ml O₂/ml blood/mmHg.

In the event that the described linear approximation is insufficientlyaccurate, such as in the case of severe pathology, it will beappreciated that, instead of approximating the dissociation curve usinga linear function it is possible to more closely approximate the curveusing a more complex function.

The combined blood and tissues (i.e. the biological material within theregion of interest) therefore has a solubility coefficient of its own(α′_(O) ₂ ) which is defined in equation 10:

$\begin{matrix}{\alpha_{O_{2}}^{\prime} = \frac{{f_{B}\left( {{CO} + {\beta_{O_{2}}{PW}_{O_{2}}}} \right)}f_{W}\alpha_{O_{2}}{PW}_{O_{2}}}{{PW}_{O_{2}}}} & (10)\end{matrix}$

where f_(B) is the fraction of blood in the space; f_(W) is the fractionof water in the space; β_(O) ₂ is a standard solubility coefficient foroxygen in blood in units of ml O₂/ml blood/mmHg and α_(O) ₂ is thesolubility coefficient of oxygen in water and PW _(O) ₂ is the partialpressure of dissolved oxygen.

Taking into account the proportion of a region of interest which isbiological material and the solubility coefficient of that biologicalmaterial, the net change in concentration of oxygen in the blood andtissue water for a given change in partial pressure PW _(O) ₂ is givenby the input from diffusion less the output from blood flow, which isexpressible as equation 3 less equation 8, as set out in equation 11:

$\begin{matrix}{{\alpha_{O_{2}}^{\prime}v_{W}\frac{{PW}_{O_{2}}}{t}} = {{{DM}_{O_{2}}\left( {{PA}_{O_{2}} - {PW}_{O_{2}}} \right)} - {\overset{.}{Q}\left( {{Cc}_{O_{2}}^{\prime} - {C{\overset{\_}{v}}_{O_{2}}}} \right)}}} & (11)\end{matrix}$

where α′_(O) ₂ is the solubility coefficient for oxygen into thebiological material in the region of interest within the lung, includingboth tissue water and blood; v_(W) is the fractional volume of theregion of interest which is biological material; and dPW _(O) ₂ is thedifference in partial pressure of oxygen in the region of interest.

Substituting equation 9 into equation 11 gives a mathematical model ofthe net change in dissolved partial pressure of oxygen in the blood andtissue water in a region of interest in terms of partial pressures, asset out in equation 12:

$\begin{matrix}{{\alpha_{O_{2}}^{\prime}v_{W}\frac{{PW}_{O_{2}}}{t}} = {{{DM}_{O_{2}}\left( {{PA}_{O_{2}} - {PW}_{O_{2}}} \right)} - {\overset{.}{Q}\left( {{CO} + {\beta_{O_{2}}{PW}_{O_{2}}} - {C{\overset{\_}{v}}_{O_{2}}}} \right)}}} & (12)\end{matrix}$

As with FI _(O) ₂ , C v _(O) ₂ can conveniently be estimated in units ofa fractional volume of oxygen to volume of blood (i.e. fractionalconcentration) and need not therefore be converted to a partialpressure. If values of C v _(O) ₂ were measured over time by some methodthen conversion to units of partial pressure may be desirable and can beachieved by performing a substitution into equation 11 using an identityof the kind set out in equation 10, albeit specific to the relationshipbetween the available measurement, the concentration of oxygen in thevenous blood C v _(O) ₂ and the partial pressure of oxygen in the venousblood.

It will be appreciated that equations 7 and 12 represent two componentsof a mathematical model 13 which characterise respectively thetransmission of oxygen from an external source which is breathed by thesubject into the alveolar spaces in the lung 1 (ventilation, {dot over(V)}I, which is described above as being is assumed to be equal to the{dot over (V)}A term in equation 7) and the transport of oxygen intodeoxygenated blood 4 in the lung 1 which is then carried away by theoxygenated blood 6 (perfusion, {dot over (Q)}). It will be appreciatedthat such a model 13 can advantageously be used to infer measurements ofphysiological parameters which cannot be measured, or are difficult tomeasure, directly. The model components 13 have a large number ofparameters, any of which may provide useful clinical informationprovided that reliable values for that parameter can be obtained. Oneway to produce values for the parameters in a model is to obtainmeasurements (over time) for a parameter or parameters to the model andfind values for the remaining parameters which (when the model isnumerically evaluated for each of the modelled values) would producevalues similar to, or the same as, those measured for the measuredparameter. Another method would be to estimate values for an estimatedparameter by some reliable method other than the mathematical model andfind values for the other parameters which would produce values similarto the estimated parameters. The method of finding appropriate valuesfor the parameters of the model can be any appropriate method; forexample a least squared error fitting algorithm such as theLevenberg-Marquardt algorithm may be used. It will be appreciated thatthe more parameters in a model which can reliably be measured orestimated, the greater the likelihood of accuracy in the modelled valuesof the other parameters.

When using the model 13 with OE-MRI data, the concentration of oxygenbreathed by the subject (either estimated as FI _(O) ₂ or measured) isan input to the ventilation model (equation 7). A further input to thatmodel is the partial pressure of dissolved oxygen in a region ofinterest within the lung tissues and blood, PW _(O) ₂ , which is inputto the model from the 4D dataset of partial pressure values produced bythe OE-MRI study described above. For example, a set of data valuesinput to the model might be l(2,3,4,t) where the (x,y,z) co-ordinates ofthe spatial location within the lungs are (2,3,4) and t represents anacquisition time. Another input to the model could be, for example, thesystemic venous concentration of oxygen C v _(O) ₂ , which could beestimated from known statistical averages for age and disease groups.Alternatively, C v _(O) ₂ could be measured directly either from theOE-MRI study (by measuring the partial pressure of oxygen in a largevein, such as the vena cava) or by monitoring a patient's blood gases byany other clinically established method.

It will be appreciated that in order to fit equation 12 to values of PW_(O) ₂ using a least squared error fitting algorithm it is firstnecessary to solve equation 12 so that the resulting equation contains aPW _(O) ₂ parameter rather than a dPW _(O) ₂ parameter. Similarly, ifvalues of PA _(O) ₂ were obtained and used as inputs to the model thenequation 7 would need to be solved so as to include a PA _(O) ₂ terminstead of a dPA _(O) ₂ term. It will be appreciated that there are manymethods of solving the equations algebraically to achieve this goal.

Although obtaining more measurements for use as inputs to the modelincreases the accuracy of the outputs from the model after fitting, itis not always desirable to obtain a number of different measurements fordifferent model parameters over time so as to feed them all into themodel. An alternative approach to increasing the accuracy of modelledparameter values resulting from a fit of the model 13 to measured datais to make assumptions about the model or its parameters in the hope ofsimplifying the models. One possible approach to this, set outimmediately below, is tailored to generate a model suitable for fittingto the OE-MRI data and other data derived from measurements and/orestimations so as to generate values of {dot over (V)}A and {dot over(Q)} (respectively ventilation, in units of ml gas/min/ml lung, andperfusion, or blood flow, in units of ml blood/min/ml lung). Theresulting values of {dot over (V)}A and {dot over (Q)} may then beanalysed individually or combined (e.g. by dividing the value for {dotover (V)}A for each region of interest by the corresponding value for{dot over (Q)}) to represent the mismatch between ventilation andperfusion for each of the regions of interest within the lungs. It willbe apparent that accurate values for such measurements would beextremely valuable to the clinician in the diagnosis, prognosis andtreatment of patients with pulmonary dysfunction, for example patientssuffering from COPD. In particular, given that {dot over (V)}Arepresents the volume of air ventilated into a region of interest withinthe lung over a given time period, and given that all of the lung isbeing imaged (as a plurality of adjacent regions of interest), if avalue can be determined for {dot over (V)}A in each region of interestand all of these values for the lungs summed together, then the totalshould be (at least approximately) equal to the volume of air breathedin by the subject over the same time period. This represents an exampleof the ways in which the results of the model 13 can be verified byrelated measurements.

Given that, in any region of interest within a lung, part of the regioncan be made up of alveolar spaces (the change in oxygen in which ismodelled in equation 7) and part made up of tissue and blood spaces (thechange in oxygen in which is modelled in equation 11), it can be assumedthat everything within the lung may be modelled by either the modelcomponent of equation 7 or by the model component of equation 11. Itwill of course be appreciated that some regions of interest may beentirely alveolar space or entirely lung tissue and blood. Thus, it maybe assumed that the fractional volume of biological material v_(W) in aregion of interest and the fractional volume of gaseous alveolar spacesv_(A) in the region of interest, when summed together, add up to one.This assumption allows equation 7 to be set out as follows in equation13:

$\begin{matrix}{{\frac{1 - v_{W}}{{PB} - P_{H_{2}0}}\frac{{PA}_{O_{2}}}{t}} = {{\overset{.}{V}{A\left( {{FI}_{O_{2}} - \frac{{PA}_{O_{2}}}{{PB} - P_{H_{2}0}}} \right)}} - {{DM}_{O_{2}}\left( {{PA}_{O_{2}} - {PW}_{O_{2}}} \right)}}} & (13)\end{matrix}$

That is the :identity v_(A)=1−v_(W) is used in equation 13 such thateverything which is not tissue and blood is considered to be alveolargas space, and this proportion of the whole may be represented as(1−v_(W)).

Recalling equation 12:

$\begin{matrix}{{\alpha_{O_{2}}^{\prime}v_{W}\frac{{PW}_{O_{2}}}{t}} = {{{DM}_{O_{2}}\left( {{PA}_{O_{2}} - {PW}_{O_{2}}} \right)} - {\overset{.}{Q}\left( {{CO} + {\beta_{O_{2}}{PW}_{O_{2}}} - {C{\overset{\_}{v}}_{O_{2}}}} \right)}}} & (12)\end{matrix}$

Equations 13 and 12 can be combined by addition so as to produce asingle equation for a model 13 of the change in the partial pressure ofdissolved oxygen in the lung 1 including the parameters of both models,as set out in equation 14:

$\begin{matrix}{{\left( {\frac{1 - v_{W}}{{PB} - P_{H_{2}0}}\frac{{PA}_{O_{2}}}{t}} \right) + \left( {v_{W}\alpha_{O_{2}}^{\prime}\frac{{PW}_{O_{2}}}{t}} \right)} = {{\overset{.}{V}{A\left( {{FI}_{O_{2}} - \frac{{PA}_{O_{2}}}{{PB} - P_{H_{2}0}}} \right)}} - {\overset{.}{Q}\left( {{CO} + {\beta_{O_{2}}{PW}_{O_{2}}} - {C{\overset{\_}{v}}_{O_{2}}}} \right)}}} & (14)\end{matrix}$

The terms for diffusion in equations 12 and 13 are equal but haveopposite sign, and they therefore cancel when equations 12 and 13 areadded together as shown in equation 14.

A useful assumption to make is that the partial pressure of oxygen inthe alveolar spaces is (at least approximately) equal to the partialpressure of oxygen in the surrounding tissue water and blood. This is areasonable approximation given that diffusion between the two spacesoccurs to correct an imbalance between the partial pressures of oxygenbetween the two spaces, and that the time scales involved in correctingthe imbalance are relatively small. The approximation may not, ofcourse, be appropriate in the case of abnormal lung function resultingin severe impairment of diffusion of oxygen between the alveolar spacesand the biological material in the lungs, e.g. in a subject sufferingfrom severe emphysema or interstitial fibrosis. However Wagner and West(JAP, 33, 62-71, 1972) have shown that the assumption holds true even ifdiffusion capacity is as low as 25% of normal diffusion capacity. Theapproximation is set out in equation 15:

Assume :P A _(O) ₂ =P W _(O) ₂   (15)

where PA _(O) ₂ represents the partial pressure of oxygen in thealveolar spaces in units of mmHg; and PW _(O) ₂ represents the partialpressure of oxygen in the tissue water and blood in units of mmHg.

Equation 14 may therefore be rewritten so the dPA _(O) ₂ term isreplaced by dPW _(O) ₂ , allowing dPW _(O) ₂ /dt to be factored out ofthe right hand side of equation 15, as set out in equation 16:

$\begin{matrix}{{\left( {\frac{1 - v_{W}}{{PB} - P_{H_{2}0}} + {v_{W}\alpha_{O_{2}}^{\prime}}} \right)\frac{{PW}_{O_{2}}}{t}} = {{\overset{.}{V}{A\left( {{FI}_{O_{2}} - \frac{{PW}_{O_{2}}}{{PB} - P_{H_{2}0}}} \right)}} - {\overset{.}{Q}\left( {{CO} + {\beta_{O_{2}}{PW}_{O_{2}}} - {C{\overset{\_}{v}}_{O_{2}}}} \right)}}} & (16)\end{matrix}$

which can be written as:

$\begin{matrix}{{\left( \frac{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}}{{PB} - P_{H_{2}0}} \right)\frac{{PW}_{O_{2}}}{t}} = {{\overset{.}{V}{A\left( {{FI}_{O_{2}} - \frac{{PW}_{O_{2}}}{{PB} - P_{H_{2}0}}} \right)}} - {\overset{.}{Q}\left( {{CO} + {\beta_{O_{2}}{PW}_{O_{2}}} - {C{\overset{\_}{v}}_{O_{2}}}} \right)}}} & (17)\end{matrix}$

where λ′=α′_(O) ₂ (P_(B)−P_(H) ₂ _(O)) is a substitution which has beenmade for ease of representation, and represents the partitioncoefficient of oxygen between the gas phase and the blood and tissuewater phase expressed in terms of barometric pressure (less water vapourpressure) multiplied by the solubility coefficient of oxygen in theblood and tissue water.

The combined model 13 of equation 17 may be fitted for values of PW _(O)₂ given by the 4D dataset produced by the OE-MRI study described abovein the same way as described in relation to equations 7 and 12 but first(as mentioned above) the differential should be solved so that the modelhas a parameter for PW _(O) ₂ . This may be achieved using anintegrating factor as described below.

Equation 17 is rearranged into a form suitable for application of anintegrating factor. Equation 17 is first rewritten by dividing bothsides by

$\left( \frac{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}}{{PB} - P_{H_{2}O}} \right)$

to give:

$\begin{matrix}{\frac{{PW}_{O_{2}}}{t} = {\frac{{PB} - P_{H_{2}O}}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}}\left( {{\overset{.}{V}{A\left( {{FI}_{O_{2}} - \frac{{PW}_{O_{2}}}{{PB} - P_{H_{2}0}}} \right)}} - {Q\left( {{CO} + {\beta_{O_{2}}{PW}_{O_{2}}} - {C{\overset{\_}{v}}_{O_{2}}}} \right)}} \right)}} & (18)\end{matrix}$

expanding the outermost bracket of the right hand side of equation 18gives:

$\begin{matrix}{\frac{{PW}_{O_{2}}}{t} = {{\frac{\overset{.}{V}{A\left( {{PB} - P_{H_{2}O}} \right)}}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}}\left( {{FI}_{O_{2}} - \frac{{PW}_{O_{2}}}{{PB} - P_{H_{2}O}}} \right)} - {\frac{Q\left( {{PB} - P_{H_{2}O}} \right)}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}}\left( {{CO} + {\beta_{O_{2}}{PW}_{O_{2}}} - {C{\overset{\_}{v}}_{0_{2}}}} \right)}}} & (19)\end{matrix}$

Further expansion gives:

$\begin{matrix}{\frac{{PW}_{O_{2}}}{t} = {\frac{\overset{.}{V}{A\left( {{PB} - P_{H_{2}O}} \right)}{FI}_{O_{2}}}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}} - \frac{\overset{.}{V}{A\left( {{PB} - P_{H_{2}O}} \right)}{PW}_{O_{2}}}{1 - {\left( {1 - \lambda^{\prime}} \right){v_{W}\left( {{PB} - P_{H_{2}O}} \right)}}} - \frac{{Q\left( {{PB} - P_{H_{2}O}} \right)}{CO}}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}} - \frac{{Q\left( {{PB} - P_{H_{2}O}} \right)}\beta_{O_{2}}{PW}_{O_{2}}}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}} + \frac{{Q\left( {{PB} - P_{H_{2}O}} \right)}C{\overset{\_}{v}}_{0_{2}}}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}}}} & (20)\end{matrix}$

Factoring based on PW_(O) ₂ and (PB−P_(H) ₂ _(O)) then gives:

$\begin{matrix}{\frac{{PW}_{O_{2}}}{t} = {{{- \left( {\frac{\overset{.}{V}A}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}} + \frac{{Q\left( {{PB} - P_{H_{2}O}} \right)}\beta_{O_{2}}}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}}} \right)}{PW}_{O_{2}}} + {\left( {{PB} - P_{H_{2}O}} \right)\left( {\frac{\overset{.}{V}{AFI}_{O_{2}}}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}} - \frac{QCO}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}} + \frac{{QC}{\overset{\_}{v}}_{0_{2}}}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}}} \right)}}} & (21)\end{matrix}$

Which can be rewritten as:

$\begin{matrix}{{\frac{{PW}_{O_{2}}}{t} + {\left( {\frac{\overset{.}{V}A}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}} + {\left( \frac{Q}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}} \right){\beta_{O_{2}}\left( {{PB} - P_{H_{2}O}} \right)}}} \right){PW}_{O_{2}}}} = {\left( {{PB} - P_{H_{2}O}} \right) + \left( {\frac{\overset{.}{V}{AFI}_{O_{2}}}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}} - {\frac{Q}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}}\left( {{C{\overset{\_}{v}}_{0_{2}}} - {CO}} \right)}} \right)}} & (22)\end{matrix}$

when v, q and λ_(B) are defined as follows:

$\begin{matrix}{v = \frac{\overset{.}{V}A}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}}} & (23) \\{q = \frac{\overset{.}{Q}}{1 - {\left( {1 - \lambda^{\prime}} \right)v_{W}}}} & (24) \\{\lambda_{B} = {\beta_{O_{2}}\left( {P_{B} - P_{H_{2}O}} \right)}} & (25)\end{matrix}$

Allows equation 22 to be rewritten as:

$\begin{matrix}{{\frac{{PW}_{O_{2}}}{t} + {\left( {v + {q\; \lambda_{B}}} \right){{PW}_{O_{2}}(t)}}} = {\left( {P_{B} - P_{H_{2}O}} \right)\left( {{{vFI}_{O_{2}}(t)} + {q\left( {{C{{\overset{\_}{v}}_{O_{2}}(t)}} - {CO}} \right)}} \right)}} & (26)\end{matrix}$

The parameters v, q and λ_(B) represent components of the model whichare constant, i.e. their values do not change in a particular region ofinterest within a subject's lungs unless the function of that region ofthe lung changes. It should be noted that equations 23 and 24 inparticular comprise a common denominator together with a parameterrespectively representing ventilation ({dot over (V)}A) and perfusion({dot over (Q)}). λ_(B) is the partition coefficient of oxygen betweenthe alveolar spaces and the blood (whereas λ′ is the partitioncoefficient of oxygen between the alveolar spaces and all of thenon-gaseous spaces in the region of interest). In equation 26, the PW_(O) ₂ , FI _(O) ₂ and C v _(O) ₂ terms comprise indices of time t. Itwill be appreciated that values of the lung parameters represented bythese terms vary over time and that the terms themselves are functionsof time. It will further be appreciated that the indices are implicit inequations 1 to 18 but have been omitted for clarity of representation.The indices are shown in equation 19 and all subsequent equationsbecause they are relevant both to the use if the integrating factor andto the fitting of the model to data values obtained for a plurality oftime points

Equation 26 is in a standard form required for the application of anintegrating factor, the standard form being set out in equation 27:

$\begin{matrix}{{{\frac{y}{x} + {{P(x)}y}} = {Q(x)}}{where}} & (27) \\{{{y(x)} = {{PW}_{O_{2}}(t)}},} & (27.1) \\{{x = t},} & (27.2) \\{{{P(x)} = {v + q_{B}}},} & (27.3) \\{{{Q(x)} = {\left( {P_{B} - P_{H_{2}O}} \right)\left( {{{vFI}_{O_{2}}(t)} + {{qC}{{\overset{\_}{v}}_{O_{2}}(t)}}} \right)}};{and}} & (27.4) \\{\frac{y}{x} = \frac{{PW}_{O_{2}}}{t}} & (27.5)\end{matrix}$

According to the method of using an integrating factor, equation 27 maybe solved by multiplying by an integrating factor e^(∫P(x)dx) andsimplifying to give equation 28:

$\begin{matrix}{\frac{{y}\; ^{\int{{P{(x)}}{x}}}}{x} = {{Q(x)}^{\int{{P{(x)}}{x}}}}} & (28)\end{matrix}$

Integrating equation 28 gives

$\begin{matrix}{{y(x)} = {\frac{1}{^{\int{{P{(x)}}{x}}}}{\int{{Q(x)}^{\;^{\int{{P{(x)}}{x}}}}{x}}}}} & (29)\end{matrix}$

Equation 19 may therefore be multiplied by an integrating factor definedas e^(∫P(x)dx)=e^(∫v+qλ) ^(B) ^(dt)=e^((v+qλ) ^(B) ^()t). Substitutingthe definitions of 27.1 to 27.5 into equation 29 gives:

P W _(O) ₂ (t)=e ^(−(v+qλ) ^(B) ^()t)∫(P _(B) −P _(H) ₂ _(O))(vF I _(O)₂ (t)+q(C v _(O) ₂ (t)−CO))e ^((v+qλ) ^(B) ^()t) dt  (30)

Equation 30 can be rewritten as a definite integral because for a giventime t the limits are known to be 0 and t, provided that a term for theinitial condition PW _(O) ₂ (0) is added to the right hand side, as setout in equation 27.

P W _(O) ₂ (t)=e ^(−(v+qλ) ^(B) ^()t)(∫₀ ^(t)[(P _(B) −P _(H) ₂ _(O))(vFI _(O) ₂ (t′)+q(C v _(O) ₂ (t′)−C O))e ^((v+qλ) ^(B) ^()t′) ]dt′+P W_(O) ₂ (0))  (31)

where t′ is an index between 0 and t within the integration. Equation 31can be rewritten as set out in equation 32:

PW _(O) ₂ (t)=∫₀ ^(t)[(P _(B) −P _(H) ₂ _(O))(vF I _(O) ₂ (t′)+q(C v_(O) ₂ (t′)−CO))e ^((v+qλ) ^(B) ^()(t′−t)]dt′+P W _(O) ₂ (0)e ^(−(v+qλ)_(B) ^()t)  (32)

By using the initial condition that

${\frac{{PW}_{O_{2}}}{t} = 0},$

given equation (26) PW _(O) ₂ (0) can be defined as set out in equation33:

$\begin{matrix}{{{PW}_{O_{2}}(0)} = {\frac{P_{B} - P_{H_{2}O}}{v + {q\; \lambda_{B}}}\left( {{{vFI}_{O_{2}}(0)} + {q\left( {{C{{\overset{\_}{v}}_{O_{2}}(0)}} - {CO}} \right)}} \right)}} & (33)\end{matrix}$

Equation 33 may readily be substituted into equation 32 so as to removeany PW _(O) ₂ terms from the left hand side of the equation. Recallingthe description of the nature of the input data from OE-MRI, the datarepresents changes from the initial conditions measured during thebaseline measurement at time t=0. It is therefore necessary to derive anequation representing the difference between PW _(O) ₂ (t) and theinitial condition PW _(O) ₂ (0), as set out in equation 34 bysubtracting equation 33 into equation 32:

$\begin{matrix}{{{P_{W_{O_{2}}}(t)} - {P_{W_{O_{2}}}(0)}} = {{\int_{0}^{t}{\left\lbrack {\left( {P_{B} - P_{H_{2}O}} \right)\left( {{{vFI}_{O_{2}}\left( t^{\prime} \right)} + {q\left( {{C{{\overset{\_}{v}}_{O_{2}}\left( t^{\prime} \right)}} - {Co}} \right)}} \right)^{{({v + {q\; \lambda_{B}}})}{({t^{\prime} - t})}}} \right\rbrack \ {t^{\prime}}}} + {\left( {\frac{P_{B} - P_{H_{2}O}}{v + {q\; \lambda_{B}}}\left( {{{vFI}_{O_{2}}(0)} + {q\left( {{C{{\overset{\_}{v}}_{O_{2}}(0)}} - {Co}} \right)}} \right)} \right)\left( {^{{- {({v + {q\; \lambda_{B}}})}}t} - 1} \right)}}} & (34)\end{matrix}$

Equation 35 can be rewritten as set out in equation 36:

$\begin{matrix}{{\frac{\Delta \; {{PW}_{O_{2}}(t)}}{P_{B} - P_{H_{2}O}} = {{v{\int_{0}^{t}{^{{({v + {q\; \lambda_{B}}})}{({t^{\prime} - t})}}\Delta \; {{FI}_{O_{2}}\left( t^{\prime} \right)}\ {t^{\prime}}}}} + {q{\int_{0}^{t}{^{{({v + {q\; \lambda_{B}}})}{({t^{\prime} - t})}}\Delta \; C{\overset{\_}{v}}_{O_{2}}\left( t^{\prime} \right)\ {t^{\prime}}}}}}}\mspace{79mu} {where}\mspace{79mu} {{{\Delta \; {{PW}_{O_{2}}(t)}} = {{{PW}_{O_{2}}(t)} - {{PW}_{O_{2}}(0)}}};}\mspace{79mu} {{{\Delta \; {{FI}_{O_{2}}(t)}} = {{{FI}_{O_{2}}(t)} - {{FI}_{O_{2}}(0)}}};{and}}\mspace{79mu} {{\Delta \; C{{\overset{\_}{v}}_{O_{2}}(t)}} = {{C{{\overset{\_}{v}}_{O_{2}}(t)}} - {C{{{\overset{\_}{v}}_{O_{2}}(0)}.}}}}} & (36)\end{matrix}$

which may be rewritten as set out in equation 35:

P W _(O) ₂ (t)−P W _(O) ₂ (0)=(P _(B) −P _(H) ₂ _(O))v∫ ₀ ^(t)(F I _(O)₂ (t′)−F I _(O) ₂ (0))e ^((v+qλ) ^(B) ^()(t′−t)) dt′+(P _(B) −P _(H) ₂_(O))q∫ ₀ ^(t)(C v _(O) ₂ (t′)−C v _(O) ₂ (0)e ^((v+qλ) ^(B) ^()(t′−t))dt′  (35)

During the fitting process the components of equation 36, which aredefined in terms of t, are those for which values are measured orestimated at a plurality of time points. ΔPW _(O) ₂ (t) is a functionwhich represents the change in partial pressure of dissolved oxygenabove baseline in the region of interest within the lung, which ispreferably measured by OE-MRI in which a scan is performed to generatedata for each time t. ΔFI _(O) ₂ (t) is a function in terms of t whichrepresents the change in fractional concentration of inspired oxygenabove baseline. This is preferably estimated (from the knownconcentrations of gases breathed by the subject) so as to produce avalue for each time t. ΔC v _(O) ₂ (t) is a function which representsthe change in partial pressure of oxygen in venous blood as it entersthe lungs above baseline. Values of ΔC v _(O) ₂ (t) may be estimatedfrom known physiological averages for each time t but may alternativelybe measured. v_(W) is the proportion of non-gaseous space to total spacein the region of interest and is preferably measured. v_(W) can bemeasured using a variety of tomographic density imaging techniquesincluding a standard MR proton density map or helical x-ray computedtomography (CT). These model parameters form the inputs to the model 13.Once the inputs have been determined, values for the constant parameterswhich make up the constants v, q and λ_(B) (i.e. {dot over (V)}A and{dot over (Q)}) and which are not standard values are determined byfitting, preferably using a least squared error approximator such as theLevenberg Marquardt algorithm. It will be appreciated that such analgorithm can operate by evaluating the equation using a variety of testvalues in place of the unknowns and calculating squared error betweenthe result of the evaluation and the measured values for one or more ofthe model parameters. The set of test values which produces a minimal,or “least”, squared error is then output from the algorithm as the bestapproximation to values for the unknown model parameters.

It will be appreciated that in order to produce independent values for{dot over (V)}A and {dot over (Q)} it is necessary to determine valuesof v_(W) for input to the model. However, if only relative values of{dot over (V)}A and {dot over (Q)} are required, such as {dot over(V)}A/{dot over (Q)}, which would be sufficient to identify, forexample, ventilation perfusion mismatch, then v_(W) need not bedetermined as an input as it cancels as a result of dividing {dot over(V)}A by {dot over (Q)}.

A further simplification of the model can be made if it is assumed thatthe value of ΔFI _(O) ₂ is constant after time t=0, i.e. if it isassumed that the change in ΔFI _(O) ₂ takes the form of a step function,that the subject goes from breathing one concentration instantaneouslyto breathing a different concentration at time t=0, and then remains atthis different concentration. Assuming that this is the case, equation36 can be rewritten because the first integral of equation 36 can beintegrated to give equation 37:

$\begin{matrix}{\frac{\Delta \; {{PW}_{O_{2}}(t)}}{P_{B} - P_{H_{2}O}} = {{\frac{v}{\left( {v + {q\; \lambda_{B}}} \right)}\Delta \; {{FI}_{O_{2}}\left( {1 - ^{{- {({v + {q\; \lambda_{B}}})}}t}} \right)}} + {q{\int_{0}^{t}{^{{({v + {q\; \lambda_{B}}})}{({t^{\prime} - t})}}\Delta \; C{{\overset{\_}{v}}_{O_{2}}\left( t^{\prime} \right)}\ {t^{\prime}}}}}}} & (37)\end{matrix}$

It is not advisable to repeat this simplification for the remainingintegral in equation 37 by assuming that ΔC v _(O) ₂ is constant becausethis parameter represents the venous concentration of oxygen, which isan important factor and will change throughout at least part of the timeof the study. However various substitutions can be made to furthersimplify the model if desired. ΔC v _(O) ₂ should therefore be measuredor estimated, at least to an approximation, as an input to the model 13so that the model 13 obtains valuable clinically acceptable values forthe parameters {dot over (V)}A and {dot over (Q)}.

In general it will be appreciated that any of the models 13 (i.e. at anystage of simplification) described with reference to equations 1 to 37may be fitted to measured data in the manner described above so as togenerate values for unknown model parameters (i.e. model parameterswhich have not been measured or estimated by some other means).

Results of application of the invention are now described with referenceto FIGS. 5 and 6. The model 13 of the first embodiment as set out inequation 33 was used in conjunction with OE-MRI studies in order tocompare the lung function of a group of healthy and unhealthy subjects.The unhealthy subjects all suffered from chronic obstructive pulmonarydisease (COPD). The healthy subjects were age matched so as to share theage range of the COPD subjects. The healthy subjects had no knownpulmonary disorder.

An OE-MRI study was performed on each subject using a Philips Intera1.5T MRI scanner. Each subject breathed medical air (21% oxygen) andthen subsequently breathed 100% oxygen, both at a flow rate of 15 l/min,using a 3-valve Hudson non-rebreathing mask. Care was taken in each caseto ensure a tight seal of the mask against the face. Existing research(Boumphrey S M, Morris E A, Kinsella S M. “100% inspired oxygen from aHudson mask—a realistic goal?” Resuscitation. April 2003;57(1):69-72.)suggests that it should be possible to achieve fractional inspiredoxygen concentrations of around 97% with such an arrangement.

For each subject in turn, while the subject breathed the medical air aset of single-slice inversion-recovery half Fourier single shot turbospin echo (IR-HASTE) scans was performed with a range of inversion times(TI=50, 300, 1100, 2000, 5000 ms) so as to produce a 3D dataset ofbaseline R₁ values. The coronal imaging slice was positioned posteriorlyand the data values were acquired for each slice using a 128×128 matrixof locations on the slice. The in-plane field of view was 450×450 mm andthe slice thickness was 10 mm. Thus, each location within the resultingdataset represents a 3.5 mm×3.5 mm×10 mm volume region within thesubject. The echo time (TE) was 3 ms and the time between pulsesequences (TR) was 5500 ms. The purpose of this first set of data was todetermine the baseline longitudinal relaxation time T1 for all locationsthroughout the lungs of the subject.

The subject was then scanned continuously with a temporal resolution(i.e. time between the start of one scan and the start of the next scan)of 5.5 s using the same IR-HASTE protocol with an inversion time set at1100 ms so as to produce further 3D datasets for each of a plurality oftimes t. After the 15^(th) scan, the gas was switched from medical airto 100% oxygen and scanning was continued for a further 6 minutes. A setof IR-HASTE scans was then performed with the same range of inversiontimes as used for the baseline scans was then acquired while the subjectcontinued to breathe 100% oxygen. The subjects were free breathingthroughout the protocol. Subjects tolerated the protocol well with noadverse events.

Prior to fitting of the model of the first embodiment to the data, animage registration algorithm was applied to the data (in accordance withthe method described in Naish J H, Parker G J, Beatty P C, Jackson A,Young S S, Waterton J C, Taylor C J. “Improved quantitative dynamicregional oxygen-enhanced pulmonary imaging using image registration.”Magnetic Resonance in Medicine. August 2005; 54(2):464-9) in order tocorrect for respiratory motion between scans. T₁-maps were generated forair and oxygen breathing by fitting on a voxel by voxel basis to astandard inversion-recovery equation. The dynamic T₁-weighted imageswere converted first to T, values using the baseline T₁ map ascalibration and then to change in partial pressure of oxygen dissolvedin the parenchymal tissue water and blood plasma ΔPW _(O) ₂ (t) usingthe relaxivity constant r₁=2.49×10⁻⁴ (Zaharchuk G, Martin A J, Dillon WP. “Noninvasive imaging of quantitative cerebral blood flow changesduring 100% oxygen inhalation using arterial spin-labeling MR imaging.”American Journal of Neuroradiology. April 2008; 29(4):663-7) asdescribed above. The mathematical model described above was then fittedto the data in the manner described above in order to extract parametersrelating to regional ventilation and perfusion and quantitativeventilation/perfusion ({dot over (V)}A/{dot over (Q)}) maps weregenerated.

FIG. 5 shows example {dot over (V)}A/{dot over (Q)} maps following imageregistration and model fitting for two subjects of approximately equalage: (a) a healthy volunteer, and (b) a subject with COPD. In thehealthy volunteer the {dot over (V)}A/{dot over (Q)} map is relativelyuniform across the lungs and in a normal range (around 0.8). In the COPDsubject, {dot over (V)}A/{dot over (Q)} is generally lower (shown bydarker areas) and appears more spatially heterogeneous (i.e. differentregions of the lungs have more widely differing {dot over (V)}A/{dotover (Q)} values than in the healthy volunteer).

FIG. 6 shows histograms (a-c) for three different subjects with COPD,and (d-f) three healthy volunteers. In each histogram the marks on thex-axis indicate values of {dot over (V)}A/{dot over (Q)} and the markson the y-axis indicate numbers of locations within the lungs. In thehealthy volunteers a narrow peak is observed which indicates that mostof the values of {dot over (V)}A/{dot over (Q)} throughout eachsubject's lungs are similar and cluster around 0.8; in the COPD subjectsa broad peak, centered at relatively lower {dot over (V)}A/{dot over(Q)}, is observed, which indicates that the values of {dot over(V)}A/{dot over (Q)} throughout the lung are more heterogeneous and thatthe lungs in general are more poorly ventilated than in the healthysubjects.

It will be appreciated from both FIGS. 5 and 6 that there is basis uponwhich to make a diagnosis of impaired lung function in the COPD subjectsin comparison with the healthy volunteers. Moreover, given that thevalues of {dot over (V)}A/{dot over (Q)} are in units of ml gas/mlblood, the values themselves represent a quantitative measurement oflung function for each area within the lungs. In general terms it willbe appreciated that there may be many clinical uses of such data. Itwill further be appreciated that the {dot over (V)}A/{dot over (Q)}values may readily be represented as individual values of {dot over(V)}A and {dot over (Q)} for each location within the lungs if a valueof v_(W) can be determined for each location.

It will generally be appreciated that the above described embodiment ismerely exemplary and is not intended to limit the scope of theinvention. In particular, it will be appreciated that the particularmeasurements, estimates and assumption used to simplify the model mightreadily be replaced by other measurements, estimates and assumptionsprovided that they are clinically acceptable and do not compromise theacceptability of the data output by the model. It will further beappreciated that the model of the function of the lung might derisablybe subdivided into different model components and, in some cases, morethan two model components.

1. A method for generating a data indicative of a lung function of asubject, the method comprising: receiving a first data which has beenobtained from the subject; and inputting said first data to a model oflung function to generate said data indicative of lung function; whereinthe model of lung function comprises a first model component modelling atransfer of a gaseous oxygen from a gaseous space within the lung to abiological material within the lung based upon a first quantitative dataindicative of oxygen content in inhaled gases and oxygen content in thebiological material and a second model component modelling a transfer ofoxygen from the lungs by an oxygenation of venous blood to create anoxygenated blood based upon a second quantitative data indicative ofoxygen content in the venous blood.
 2. A method according to claim 1,wherein the second model component comprises a first parameterrepresenting a volume of blood flow.
 3. A method according to claim 2wherein the generated data includes at least one value for said firstparameter representing a volume of blood flow.
 4. A method accordingclaim 2 wherein the first model component comprises a second parameterrepresenting a volume of inhaled gases in at least part of the gaseousspace.
 5. A method according to claim 4 wherein the generated dataincludes at least one value for said second parameter representing avolume of inhaled gases in at least part of the gaseous space.
 6. Amethod according to claim 4, further comprising generating data basedupon said first parameter and second parameter.
 7. A method according toclaim 6, wherein generating data based upon said first parameter andsaid second parameter comprises performing an arithmetic operation on avalue of said first parameter and a value of said second parameter.
 8. Amethod according to claim 1, wherein said model of lung function modelslung function in a part of a lung comprising a first portion comprisingsaid gaseous space and a second portion comprising said biologicalmaterial.
 9. A method according to claim 8, wherein the second modelcomponent comprises a first parameter representing a volume of bloodflow, and said first parameter represents a volume of blood flowthorough said second portion of said part of the lung.
 10. A methodaccording to claim 8, wherein the first model component comprises asecond parameter representing a volume of inhaled gases in at least partof the gaseous space, and said second parameter represents the volume ofinhaled gasses in said first portion of said part of the lung.
 11. Amethod according to claim 1 wherein the second quantitative dataindicative of oxygen content in the venous blood comprises a pluralityof values each relating to oxygen content at a respective time.
 12. Amethod according to claim 1, wherein the method further comprisesreceiving the second quantitative data indicative of oxygen content inthe venous blood, the second quantitative data indicative of oxygencontent in the venous blood having been obtained from the subject.
 13. Amethod according to claim 1, wherein the model includes a parameterrepresenting a solubility of oxygen in blood.
 14. A method according toclaim 1, wherein the first model component comprises a first partrepresenting an amount of inhaled gases.
 15. A method according to claim14, wherein the first model component further comprises a second partrepresenting an amount of oxygen diffused into the biological materialfrom the gaseous space.
 16. A method according to claim 1, wherein thesecond model component comprises a first part representing the amount ofoxygen diffused into the biological material from the gaseous space. 17.A method according to claim 16, wherein the second model componentfurther comprises a second part representing the transfer of oxygen fromthe lungs by oxygenation of venous blood.
 18. A method according toclaim 1, wherein the model is based upon an assumption thatconcentrations of oxygen in the gaseous space within the lung issubstantially equal to a concentration of oxygen in the biologicalmaterial.
 19. A method according to claim 1, wherein the modelapproximates saturation of oxygen in the blood using a linear function.20. A method according to claim 8, wherein said model comprises a thirdparameter indicating a proportion of said part of the lung made up ofone of said first and second portions.
 21. A method according to claim20 further comprising receiving at least one value for the thirdparameter as an input.
 22. A method according to claim 21, wherein theat least one value for the third parameter has been obtained from thesubject.
 23. A method according to claim 20, wherein said first andsecond components are combined together in the model according to saidproportion.
 24. A method according to claim 8 wherein the model has theform:${\left( {\frac{\left( {1 - v_{W}} \right)}{{PB} - P_{H_{2}0}} + {v_{W}\alpha_{O_{2}}^{\prime}}} \right)\frac{{PW}_{O_{2}}}{t}} = {{\overset{.}{V}{A\left( {{FI}_{O_{2}} - \frac{{PW}_{O_{2}}}{{PB} - P_{H_{2}0}}} \right)}} - {\overset{.}{Q}\left( {{CO} + {\beta_{O_{2}}{PW}_{O_{2}}} - {C{\overset{\_}{v}}_{O_{2}}}} \right)}}$wherein PW _(O) ₂ is an input parameter representing the partialpressure of oxygen in the biological material in said part of the lung;FI _(O) ₂ is an input parameter representing the concentration of oxygeninhaled by the subject; C v _(O) ₂ is an input parameter representingthe concentration of oxygen present in the venous system of the subject;α′_(O) ₂ is an input parameter which represents the solubility of oxygenin the biological material; β_(O) ₂ is an input parameter whichrepresents the solubility coefficient of oxygen in the blood; CO is ascalar value; v_(W) is a parameter representing the proportion ofbiological material in said part of the lung, {dot over (V)}A is anoutput parameter for the model which represents the volume of inhaledgases within said gaseous space in said part of the lung; and {dot over(Q)} is an output parameter for the model which represents the volume ofblood which passes through said part of the lung.
 25. A method accordingto claim 1, wherein said first quantitative data indicative of oxygencontent in the biological material is magnetic resonance data.
 26. Amethod according to claim 1, wherein said generating of data comprisesapplication of the Levenberg Marquardt non-linear least squares fittingalgorithm to the model so as to fit said model to said input data.
 27. Amethod for generating a data indicative of a ventilation and a perfusionwithin a subject's lung the method comprising: obtaining a first dataindicative of ventilation and perfusion while the subject inhales gasescomprising a first concentration of oxygen; obtaining a second dataindicative of ventilation and perfusion while the subject inhales gasescomprising a second concentration of oxygen; generating a third databased upon the first and second data; determining a quantitative dataindicative of ventilation and perfusion based upon the third data.
 28. Amethod according to claim 27 wherein the first concentration of oxygenis approximately 21% and/or the second concentration of oxygen isapproximately 100%.
 29. A method according to claim 27, wherein thefirst and/or second data is magnetic resonance data obtained from thesubject.
 30. A method according to claim 27, wherein determiningquantitative data comprises processing the third data using amathematical model.
 31. A method according to claim 30 wherein theprocessing comprises determining at least one value of a parameter ofthe mathematical model based upon said third data.
 32. A methodaccording to claim 30 wherein the mathematical model comprises a firstmodel component modelling transfer of a gaseous oxygen from a gaseousspace within the lung to a biological material within the lung basedupon a first quantitative data indicative of oxygen content in inhaledgases and oxygen content in the biological material and a second modelcomponent modelling a transfer of oxygen from the lungs by anoxygenation of venous blood to create an oxygenated blood based upon asecond quantitative data indicative of oxygen content in the venousblood.
 33. (canceled)
 34. An apparatus for generating a data indicativeof a lung function of a subject, the apparatus comprising: means forreceiving a first data which has been obtained from the subject; andmeans for inputting said first data to a model of lung function togenerate said data indicative of lung function; wherein the model oflung function comprises a first model component modelling a transfer ofa gaseous oxygen from a gaseous space within the lung to a biologicalmaterial within the lung based upon a first quantitative data indicativeof oxygen content in inhaled gases and oxygen content in the biologicalmaterial and a second model component modelling a transfer of oxygenfrom the lungs by an oxygenation of venous blood to create an oxygenatedblood based upon a second quantitative data indicative of oxygen contentin the venous blood.
 35. An apparatus for generating a data indicativeof a ventilation and a perfusion within a subject's lung the apparatuscomprising: means for obtaining a first data indicative of ventilationand perfusion while the subject inhales gases comprising a firstconcentration of oxygen; means for obtaining a second data indicative ofventilation and perfusion while the subject inhales gases comprising asecond concentration of oxygen; means for generating a third data basedupon the first and second data; means for determining a quantitativedata indicative of ventilation and perfusion based upon the third data.36. A computer program adapted to implement a method for generating adata indicative of a lung function of a subject, the computer programcarried by a computer readable medium, the computer Program comprising:computer readable instructions arranged to cause a computer to receive afirst data which has been obtained from the subject; and input saidfirst data to a model of lung function to generate said data indicativeof lung function; wherein the model of lung function comprises a firstmodel component modelling a transfer of a gaseous oxygen from a gaseousspace within the lung to a biological material within the lung basedupon a first quantitative data indicative of oxygen content in inhaledgases and oxygen content in the biological material and a second modelcomponent modelling a transfer of oxygen from the lungs by anoxygenation of venous blood to create an oxygenated blood based upon asecond quantitative data indicative of oxygen content in the venousblood.
 37. (canceled)
 38. An apparatus for generating a data indicativeof a lung function of a subject, the apparatus comprising: a memorystoring processor readable instructions; and a processor arranged toread and execute instructions stored in said program memory; whereinsaid processor readable instructions comprise instructions arranged tocause the processor to. receive a first data which has been obtainedfrom the subject; and input said first data to a model of lung functionto generate said data indicative of lung function; wherein the model oflung function comprises a first model component modelling a transfer ofa gaseous oxygen from a gaseous space within the lung to a biologicalmaterial within the lung based upon a first quantitative data indicativeof oxygen content in inhaled gases and oxygen content in the biologicalmaterial and a second model component modelling a transfer of oxygenfrom the lungs by an oxygenation of venous blood to create an oxygenatedblood based upon a second quantitative data indicative of oxygen contentin the venous blood.